3.2.1
Preliminary Definitions and Assumptions
**
***Printout*

*
Knowing is not enough; we must apply.
Willing is not enough; we must do.
—*

** Definition.** A

** Definition.** A mapping

** Definition.** A mapping

** Definition.** A

** Definition. **A

** Definition.** A nonempty set

- If
*A*and*B*are in*G*, then*A*B*is in*G*. (The set is*closed*under the operation,*closure*.) - There exists an element
*I*in*G*such that for every element*A*in*G*,*I*A*=*A*I*=*A*. (The set has an*identity*.) - For every element
*A*in*G*, there is an element*B*in*G*such that*A*B*=*B*A*=*I*, denoted*A*^{–1}. (Every element has an*inverse.*) - If
*A, B,*and*C*are in*G*, then (*A*B*)**C*=*A**(*B*C*). (*associativity*)

*Theorem 3.0. The set of transformations of a plane is a
group under composition.*

*Proof. *The result follows from the following:

The composition of two transformations of a plane is a
transformation (Exercise 3.4).

The inverse of a transformation is a transformation
(Exercise 3.5).

The identity function is a transformation and composition of functions is
associative (Exercise 3.5).//

** Exercise 3.2. **Which of the following mappings are transformations?
Justify.

a. such that .

b. such that .

c. such that .

d. such that .

e. Let

Pbe a point in a planeS. Define byf(P) =Pand for any point ,f(Q) is the midpoint of .

** Exercise 3.3. **Let
and be transformations defined respectively by

a. Find the composition .

b. Find the composition .

c.
Find the inverse of *f*,
*f* ^{
MPSetChAttrs('ch0002','ch1',[[4,1,-2,0,0],[5,1,-2,0,0],[7,1,-3,0,0],[],[],[],[16,1,-7,0,0]])
MPInlineChar(2)
1}.

d.
Find the inverse of *g*,
*g* ^{
MPSetChAttrs('ch0003','ch1',[[4,1,-2,0,0],[5,1,-2,0,0],[7,1,-3,0,0],[],[],[],[16,1,-7,0,0]])
MPInlineChar(2)
1}.

** Exercise
3.4.** Prove the composition of two
transformations of a plane is a transformation of the plane.

** Exercise 3.5.** (a) Prove the
identity function is a transformation. (b) Prove the inverse of a
transformation of a plane is a transformation of the plane. (c) Prove the
composition of functions is associative.

3.1.2 Historical Overview 3.2.2 An Analytic Model for the Euclidean Plane |

© Copyright 2005, 2006 - Timothy Peil |